Question

Solve the following simultaneous equation.

$$\frac{2}{x}-\frac{3}{y}=15;\frac{8}{x}+\frac{5}{y}=77$$

Answer 1

$$\frac{2}{x}-\frac{3}{y}=15;\frac{8}{x}+\frac{5}{y}=77$$

$$\text{ Let }\frac{1}{x}=u\text{ and }\frac{1}{y}=v$$

So, the equation becomes

$$2u-3v=15.....\left(I\right)$$

$$8u+5v=77.....\left(II\right)$$

Multiply (I) with 4 we get

$$8u-12v=60.....\left(III\right)$$

(II) − (III)

$$8u-8u+5v-(-12v)=77-60$$

$$\Rightarrow 17v=17$$

$$\Rightarrow v=1$$

$$\text{ Putting the value of v in }\left(I\right)$$

$$2u-3\left(1\right)=15$$

$$\Rightarrow 2u=15+3=18$$

$$\Rightarrow u=9$$

Thus, 

$$\frac{1}{x}=u=9$$

$$\Rightarrow x=\frac{1}{9}$$

$$\frac{1}{y}=v=1$$

$$\Rightarrow y=1$$

$$(x,y)=(\frac{1}{9},1)$$